Problems with understanding the speed of light when emitted from a moving object

Question

So I'm having some problems understanding how if a moving object emits light, the speed of that light is still the constant, $c$, speed of light. So according to Einstein's theory of relativity, light travels the at the same speed for all inertial frames of reference. So from what I understand if a driver driving a car at a constant velocity with headlights on sees the light, it would be travelling at $c$, and an observer on the side of the road would also see the light as speed $c$. At school we have learn about time dilation and length contraction however I do not see how they could be applied in this situation to explain why the light "slows down". I would appreciate it if your answers could try to remain with a knowledge level of a high schooler.

I have also looked at other threads on this exchange, but either I do not understand them or they don't answer the question I have.

Answer 1

At school we have learn about time dilation and length contraction however I do not see how they could be applied in this situation to explain why the light "slows down".

This is a little backwards from the usual approach. Usually, we postulate the principle of relativity and the invariance of c and then use that to explain time dilation, length contraction, and the relativity of simultaneity. In principle, it would be possible to instead postulate time dilation, length contraction, and the relativity of simultaneity and use those to derive the usual postulates*, but in practice that is not typically done.

Instead, if you are having trouble with the invariance of the speed of light, you might prefer an approach that does not use it. It turns out that, if you use only the principle of relativity, you can prove that there are two possibilities. Either there is no finite invariant speed, in which case you get Galilean relativity, or there is a finite invariant speed, in which case you get special relativity. Then it becomes a simple matter of experimentally testing to see which of the two possibilities corresponds to reality.

*If you really wanted to derive the invariance of c from time dilation, length contraction, and the relativity of simultaneity, the procedure is straight forward. Time dilation, length contraction, and the relativity of simultaneity together give you the Lorentz transform:

$t’=\gamma t - \gamma v x/c^2$

$x’= \gamma x - \gamma v t$

Then a light pulse in the unprimed frame is given by

$x=ct$

so substituting in we get the equation for a light pulse in the unprimed frame

$t’=\gamma t - \gamma v t /c$

$x’=\gamma c t - \gamma v t = c t’$

So the speed of the light pulse is c in the primed frame also. The speed of light is therefore invariant.

Answer 2

The second postulate of Special Relativity states that "The speed of light in vacuum, c, is a constant, independent of the relative motion of the source". The phenomenons of length contraction and time dilation are consequences of this postulate, so is the postulate of constancy of $c$ that explains these phenomenons, not the opposite. There are two motivations for this postulate, one experimental and one theoretical.

The experimental evidence for constancy of $c$ comes from the famous Mihchelson- Morley experiment, as you can check here: https://en.wikipedia.org/wiki/Michelson–Morley_experiment.

The theoretical motivation comes from the Maxwell wave equation, which was the motivation used by Einstein to write his first article on relativity published in 1905, whose title is "On the Electrodynamics of Moving Bodies". You can check more about these topics here

Maxwell Wave Equation: https://en.wikipedia.org/wiki/Electromagnetic_wave_equation

Special Relativity Postulates: https://en.wikipedia.org/wiki/Postulates_of_special_relativity

Answer 3

Banesh Hoffmann, Einstein's collaborator, admits that, originally ("without recourse to contracting lengths, local time, or Lorentz transformations"), the Michelson-Morley experiment directly proved Newton's variable speed of light and disproved the constant speed of light:

"Moreover, if light consists of particles, as Einstein had suggested in his paper submitted just thirteen weeks before this one, the second principle seems absurd: A stone thrown from a speeding train can do far more damage than one thrown from a train at rest; the speed of the particle is not independent of the motion of the object emitting it. And if we take light to consist of particles and assume that these particles obey Newton's laws, they will conform to Newtonian relativity and thus automatically account for the null result of the Michelson-Morley experiment without recourse to contracting lengths, local time, or Lorentz transformations. Yet, as we have seen, Einstein resisted the temptation to account for the null result in terms of particles of light and simple, familiar Newtonian ideas, and introduced as his second postulate something that was more or less obvious when thought of in terms of waves in an ether." Banesh Hoffmann, Relativity and Its Roots, p.92 https://www.amazon.com/Relativity-Its-Roots-Banesh-Hoffmann/dp/0486406768

Even Wikipedia, surprisingly, tells the truth about the Michelson-Morley experiment and the speed of light:

"Emission theory, also called emitter theory or ballistic theory of light, was a competing theory for the special theory of relativity, explaining the results of the Michelson–Morley experiment of 1887. [...] The name most often associated with emission theory is Isaac Newton. In his corpuscular theory Newton visualized light "corpuscles" being thrown off from hot bodies at a nominal speed of c with respect to the emitting object, and obeying the usual laws of Newtonian mechanics, and we then expect light to be moving towards us with a speed that is offset by the speed of the distant emitter (c ± v)." https://en.wikipedia.org/wiki/Emission_theory